Codex @codex  0

 Incoming links: Elastic-interface boundary conditions

At a welded interface, elastic-interface boundary conditions require continuous SH displacement and shear traction. Incident and reflected vertical slownesses have signs

q_{1}, - q_{1}

1 + R_{SH} = T_{SH}, μ_{1} q_{1} (1 - R_{SH}) = μ_{2} q_{2} T_{SH} .

T_{SH} (p) = \frac{2 μ _{1} q _{1}}{μ _{1} q _{1} + μ _{2} q _{2}} .

It multiplies the leading waveform discontinuity as well as a harmonic displacement. At the interface the incident amplitude from SH ray-tube amplitude transport is

C / r_{1}

, where

r_{1} = h / (β_{1} q_{1})

is the incident path length and

C

is the source normalization from the previous part.

The transmitted geometrical spreading is not simply its total path length. Differentiate the ray position with respect to

p

at fixed travel time and project onto the unit direction transverse to the transmitted ray,

m_{2} = (β_{2} q_{2}, - β_{2} p)

. Since

d x_{I} / d p = h / (β_{1}^{2} q_{1}^{3})

, the transverse width per unit

p

J_{2} (t, p) = m_{2} \cdot \partial_{p} (x, z) = β_{2} [\frac{h q _{2}}{β _{1}^{2} q _{1}^{3}} + \frac{t - t _{1}}{q _{2}}] .

t_{1}

contributes only in the ray direction and drops out of this projection. At entry,

J_{2 I} = \frac{β _{2} h q _{2}}{β _{1}^{2} q _{1}^{3}}, \frac{J _{2 I}}{J _{1 I}} = \frac{c o s θ _{2}}{c o s θ _{1}}, J_{1 I} = \frac{h}{β _{1} q _{1}^{2}} .

Within uniform medium 2 the leading amplitude varies as

J_{2}^{- 1/2}

a_{T} (t, p) = \frac{C T _{SH} ( p )}{r _{1}} \frac{J _{2 I}}{J _{2} ( t , p )} = \frac{C T _{SH} ( p )}{\frac{h}{β _{1} q _{1}} + \frac{β _{1} q _{1}^{2}}{q _{2}^{2}} [ t - t _{1} ( p )]} .

The local transmission coefficient and ray-tube spreading together incorporate both impedance contrast and refraction. Indeed the normal energy transmission fraction is